1) Weeklies offer greater flexibility for trade duration. As a premium seller, I regard the sweet spot for duration around 45 days for nonearnings plays. Weeklies allow me to get nearer that sweet spot because I have more possible expiries available than just the monthlies.
2) In the vast majority of cases, using monthlies for plays is not ideal. Most of my plays are for expiries the same week as the announcement or the week immediately thereafter. Depending on when the announcement lies in relation to the monthly expiry, this could require a play setup that expires in 30 days, when I want a setup way shorter than that to take advantage of the immediate collapse that most underlyings undergo post-earnings.
3) Weeklies' strike prices are sometimes more "precise" than monthlies. Weeklies often offer .50 strike widths, which is particularly useful when you want to be somewhat surgical with your setup or the price of the underlying is such that 1.00 wide strike widths are cumbersome.
4) Weeklies are more helpful in short-term rolling scenarios where you only want to roll for an additional week to allow the trade time to work out.
So, when "shopping around" for options plays, always look to see whether the underlying offers weeklies ... . If it doesn't, I would probably pass on the vast majority of "monthly only" underlyings ... .